Question

In what ratio is the line segment joining the points (4,6) and (-7,-1) is divided by X-axis?
A.1:6
B.6:2
C.2:6
D.6:1

Answer 1

Hint: Two points are given and their ratio is asked, hence we are going to use the section formula to find out the ratio in which the line is divided.
Formula used: $x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$
                           $y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$
In the given formula above we use the terms $m,n,x,y,{x_1},{x_2},{y_1},{y_2}$. We get $m,n$from the ratio given i.e. $m:n = $the ratio we are given within the question. $x,y$are the points which we will find by using the section formula or these are the points by using which we can find the ratio too.${x_1},{x_2},{y_1},{y_2}$are the coordinates of the points given in the question. We will get the value of these points when we compare $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$with the given coordinates in the questions. So in this question we will initialize the problem with these formulas.

Complete step-by-step solution:
Coordinates given, $A\left( {4,6} \right)$and $B\left( { - 7, - 1} \right)$
Since the line is intersected by x-axis therefore let the coordinate of the point at x-axis be $\left( {x,0} \right)$.
Let ratio be $m:n$
Using section formula and substituting value in it , we get
$x = \dfrac{{ - 7m + 4n}}{{m + n}}$$,$ $y = \dfrac{{ - m + 6n}}{{m + n}}$
Equating $\left( {x,y} \right)$with the point of intersection i.e. $\left( {x,0} \right)$
$\therefore x = \dfrac{{ - 7m + 4n}}{{m + n}}$
And, $0 = \dfrac{{ - m + 6n}}{{m + n}}$
Doing cross multiplication, we get
$
   - m = - 6n \\
   \Rightarrow m = 6n \\
 $
$ \Rightarrow \dfrac{m}{n} = \dfrac{6}{1}$
$\therefore m:n = 6:1$
Which is the required answer.
Hence option (C) is correct.

Note: Substitution of values should be done carefully otherwise it should lead to incorrect solutions.Comparison of given coordinates with the solved coordinates should be equated properly.